#8324273

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(2,6)(7,9)(8,13)(14,15)(17,18)(19,21)(20,28)(22,24)(23,27)(25,31)(26,32)(29,30)(33,34,38,39)(35,37,41,47)(36,40,45,48)(42,46,43,44);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,24)(22,27)(25,30)(26,31)(28,32)(33,35)(34,37)(36,43)(38,41)(39,47)(40,44)(42,45)(46,48);(4,12)(8,13)(10,16)(14,15)(18,24)(21,23)(25,28)(30,32)(34,39)(35,41)(40,48)(42,43);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,22,31)(18,25,24,28)(19,26,27,29)(21,30,23,32)(33,36,38,45)(34,40,39,48)(35,42,41,43)(37,46,47,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,23)(20,29)(21,24)(22,27)(25,30)(26,31)(28,32)(33,37)(34,35)(36,44)(38,47)(39,41)(40,43)(42,48)(45,46);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,23)(18,19,24,27)(20,30,31,32)(25,26,28,29)(33,35)(34,37)(36,42)(38,41)(39,47)(40,46)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=i,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8231469);